In this article, we describe some aspects of the diffuse interface modelling of incompressible flows, composed of three immiscible components, without phase change. In the diffuse interface methods, system evolution is driven by the minimisation of a free energy. The originality of our approach, derived from the Cahn-Hilliard model, comes from the particular form of energy we proposed in Boyer and Lapuerta (M2AN Math Model Numer Anal, 40:653-987,2006), which, among other interesting properties, ensures consistency with the two-phase model. The modelling of three-phase flows is further completed by coupling the Cahn-Hilliard system and the Navier-Stokes equations where surface tensions are taken into account through volume capillary forces. These equations are discretized in time and space paying attention to the fact that most of the main properties of the original model (volume conservation and energy estimate) have to be maintained at the discrete level. An adaptive refinement method is finally used to obtain an accurate resolution of very thin moving internal layers, while limiting the total number of cells in the grids all along the simulation. Different 123 464 F. Boyer et al.numerical results are given, from the validation case of the lens spreading between two phases (contact angles and pressure jumps), to the study of mass transfer through a liquid/liquid interface crossed by a single rising gas bubble. The numerical applications are performed with large ratio between densities and viscosities and three different surface tensions.

The dynamics of isolated air bubbles crossing the horizontal interface separating two Newtonian immiscible liquids initially at rest are studied both experimentally and computationally. High-speed video imaging is used to obtain a detailed evolution of the various interfaces involved in the system. The size of the bubbles and the viscosity contrast between the two liquids are varied by more than one and four orders of magnitude, respectively, making it possible to obtain bubble shapes ranging from spherical to toroidal. A variety of flow regimes is observed, including that of small bubbles remaining trapped at the fluid-fluid interface in a film-drainage configuration. In most cases, the bubble succeeds in crossing the interface without being stopped near its undisturbed position and, during a certain period of time, tows a significant column of lower fluid which sometimes exhibits a complex dynamics as it lengthens in the upper fluid. Direct numerical simulations of several selected experimental situations are performed with a code employing a volume-of-fluid type formulation of the incompressible Navier-Stokes equations. Comparisons between experimental and numerical results confirm the reliability of the computational approach in most situations but also points out the need for improvements to capture some subtle but important physical processes, most notably those related to film drainage. Influence of the physical parameters highlighted by experiments and computations, especially that of the density and viscosity contrasts between the two fluids and of the various interfacial tensions, is discussed and analysed in the light of simple models and available theories.

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Abstract. We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e., in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem. An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh. Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms. Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L 2 norm.Mathematics Subject Classification. 65N12, 65N15, 65N30, 76D05, 76D07, 76M25.

Cet article présente d'une part une stratégie de modélisation dédiée à la simulation micromécanique des interactions entre corps, et d'autre part, sa mise en oeuvre numérique. Cette stratégie repose sur une formulation de type décomposition de domaines d'une méthode multicorps périodique dans le cadre de l'approche Non Smooth Contact Dynamics de Moreau (Moreau, 1988). Les potentialités de cette méthode sont illustrées par la complexité des interactions possibles: interactions entre éléments d'une discrétisation, entre corps discrétisés ou rigides, en compression (contact) lente ou sous impact, en glissement (frottement) ou en traction (fissuration-rupture), etc. La plateforme numérique associée, Xper, repose sur une architecture orientée objet composée de bibliothèques indépendantes spécifiquement développées pour: (1) la résolution des équations aux dérivées partielles, (2) la gestion des interactions surfaciques complexes et l'intégration en temps associée, (3) la modélisation des comportements volumiques. La pertinence numérique de l'approche est illustrée sur des exemples de fissuration de matériaux hétérogènes. ABSTRACT. This paper presents a micromechanical modeling strategy for complex multibody interactions and the associated numerical framework. The strategy rests on a periodic multibody method in the framework of the Non Smooth Contact Dynamics approach of Moreau (Moreau, 1988) extended to classical domain decomposition problems. Many complex interactions can be taken into account : interactions between discrete elements, between discrete or rigid bodies, (quasistatic) contact or impact, friction or adhesion, decohesion (cracking), etc. The associated numerical platform, Xper, is composed of three independant libraries with Object Oriented Programming. The libraries are specifically developed for : (1) the solution of systems of partial differential equations (PDEs), (2) the modeling of complex interaction problems and the time discretization, (3) the integration of complex non linear constitutive models. The ability of this computational approach is illustrated by two examples of fracture of heterogeneous materials.